Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space

We study the pointwise decay estimates for the Schrödinger and wave equations on a product cone $(X,g)$, where the metric $g=d{r}^2+r^{2}h$ and $X=C(Y)=(0,\mathrm{\infty })\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. Under the assumption that the conjugate radius $\epsilon$ of $Y$ satisfies $\epsilon >\pi$, we prove the pointwise dispersive estimates for the Schrödinger and half-wave propagator in this setting. The key ingredient is the modified Hadamard parametrix on $Y$ in which the role of the conjugate points does not come to play if $\epsilon >\pi$. In a work in progress, we will further study the case that $\epsilon \le \pi$ in which the role of conjugate points come. A new finding is that a threshold of the conjugate radius of $Y$ for $L^p$-estimates in this setting is the magical number $\pi$.